\section{向量组的等价}
	
	\begin{titwo}
		已知向量组(\Rmnum{1}) $\bm \alpha_{1},\bm \alpha_{2},\bm \alpha_{3},\bm \alpha_{4}$ 线性无关，则与(\Rmnum{1})等价的向量组是\kuo.

		\onech{$\bm \alpha_{1} + \bm \alpha_{2}, \bm \alpha_{2} + \bm \alpha_{3}, \bm \alpha_{3} + \bm \alpha_{4}, \bm \alpha_{4} + \bm \alpha_{1}$}{$\bm \alpha_{1} - \bm \alpha_{2}, \bm \alpha_{2} - \bm \alpha_{3}, \bm \alpha_{3} - \bm \alpha_{4}, \bm \alpha_{4} - \bm \alpha_{1}$}{$\bm \alpha_{1} + \bm \alpha_{2}, \bm \alpha_{2} - \bm \alpha_{3}, \bm \alpha_{3} + \bm \alpha_{4}, \bm \alpha_{4} - \bm \alpha_{1}$}{$\bm \alpha_{1} + \bm \alpha_{2}, \bm \alpha_{2} - \bm \alpha_{3}, \bm \alpha_{3} - \bm \alpha_{4}, \bm \alpha_{4} - \bm \alpha_{1}$}
	\end{titwo}

	\begin{titwo}
		已知向量组(\Rmnum{1})与向量组(\Rmnum{2})，若(\Rmnum{1})可由(\Rmnum{2})线性表示，且 $r(\text{\Rmnum{1}}) = r(\text{\Rmnum{2}}) = r$. 证明：(\Rmnum{1})与(\Rmnum{2})等价.
	\end{titwo}

	\begin{titwo}
		设 $n$ 维列向量组 $\bm \alpha_{1},\bm \alpha_{2},\cdots,\bm \alpha_{m}(m < n)$ 线性无关，则 $n$ 维列向量组 $\bm \beta_{1},\bm \beta_{2},\cdots,\bm \beta_{m}$ 线性无关的充分必要条件为\kuo.

		\onech{向量组 $\bm \alpha_{1},\bm \alpha_{2},\cdots,\bm \alpha_{m}$ 可由向量组 $\bm \beta_{1},\bm \beta_{2},\cdots,\bm \beta_{m}$ 线性表出}{向量组 $\bm \beta_{1},\bm \beta_{2},\cdots,\bm \beta_{m}$ 可由向量组 $\bm \alpha_{1},\bm \alpha_{2},\cdots,\bm \alpha_{m}$ 线性表出}{向量组 $\bm \alpha_{1},\bm \alpha_{2},\cdots,\bm \alpha_{m}$ 与向量组 $\bm \beta_{1},\bm \beta_{2},\cdots,\bm \beta_{m}$ 等价}{矩阵 $\bm A = [\bm \alpha_{1},\bm \alpha_{2},\cdots,\bm \alpha_{m}]$ 与矩阵 $\bm B = \bigl[\bm \beta_{1},$ $\bm \beta_{2},$ $\cdots,$ $\bm \beta_{m}\bigr]$ 等价}
	\end{titwo}